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I have some questions about two statements from Bosch's "Algebraic Geometry and Commutative Algebra" about algebraic varieties (page 479):

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W fix an abelian variety $A$ over field $K$. Here the definition

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Following two questions:

  1. Why is for a $K$-group scheme of finite type smooth equivalent to geometrically reduced?

  2. Why under same conditions as in 1. (so $K$-group scheme of finite type) the property irreducible is equivlaent to geometrically irreducible?

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